New York City will expand its free prekindergarten program to include 3-year-olds, an initiative Mayor Bill de Blasio said will give thousands of children a better start and help low-income parents struggling to pay for child care.

Starting in September, the city will offer prekindergarten classes in two high-poverty school districts. By 2021, it would serve an estimated 62,000 3-year-olds and cost about $877 million a year. The...